In subsequent lectures we will use what we have learned here to
extrapolate back to some of the earliest times in the universe. We will
discuss the thermodynamics of the early universe, and the resulting
interdependency between particle physics and cosmology. However, before
that, we would like to explore some implications for the future of the
universe.

For a long time in cosmology, it was quite commonplace to refer to the
three possible geometries consistent with homogeneity and isotropy as
closed (k = 1), open (k = - 1) and flat (k =
0). There were two reasons for this. First, if one considered only the
universal covering spaces, then a positively curved universe would be a
3-sphere, which has finite volume and hence is closed, while a
negatively curved universe would be the hyperbolic 3-manifold
3, which has
infinite volume and hence is open.

Second, with dust and radiation as sources of energy density, universes
with greater than the critical density would ultimately collapse, while
those with less than the critical density would expand forever, with
flat universes lying on the border between the two. for the case of pure
dust-filled universes this is easily seen from (40) and (44).

As we have already mentioned, GR is a local theory, so the first of
these points was never really valid. For example, there exist perfectly
good compact hyperbolic manifolds, of finite volume, which are
consistent with all our cosmological assumptions. However, the
connection between geometry and destiny implied by the second point
above was quite reasonable as long as dust and radiation were the only
types of energy density relevant in the late universe.

In recent years it has become clear that the dominant component of energy
density in the present universe is neither dust nor radiation, but rather
is dark energy. This component is characterized by an equation of state
parameter w < - 1/3. We will have a lot more to say about this
component (including the observational evidence for it) in the next lecture,
but for now we would just like to focus on the way in which it has
completely separated our concepts of geometry and destiny.

For simplicity, let's focus on what happens if the only energy density
in the universe is a cosmological constant, with w = - 1. In
this case, the Friedmann equation may be solved for any value of the
spatial curvature parameter k. If
> 0 then the
solutions are

(50)

where we have encountered the k = 0 case earlier. It is
immediately clear that, in the
t limit, all solutions
expand exponentially, independently of the spatial curvature. In fact,
these solutions are all exactly the same spacetime - de Sitter
space - just in different coordinate systems. These features of de
Sitter space will resurface crucially when we discuss
inflation. However, the point here is that the universe clearly expands
forever in these spacetimes, irrespective of the value of the spatial
curvature. Note, however, that not all of the solutions in (50) actually
cover all of de Sitter space; the k = 0 and k = - 1
solutions represent coordinate patches which only cover part of the
manifold.

For completeness, let us complete the description of spaces with a
cosmological constant by considering the case
< 0. This
spacetime is called Anti-de Sitter space (AdS) and it should be
clear from the Friedmann equation that such a spacetime can only exist
in a space with spatial curvature k = - 1. The corresponding
solution for the scale factor is

(51)

Once again, this solution does not cover all of AdS; for a more complete
discussion, see
[20].